Axisymmetric free vibration of thick annular plates

This paper presents a numerical analysis of the axisymmetric free vibration of moderately thick annular plates using the differential quadrature method (DQM). The plates are described by Mindlin’s first-order shear-deformation theory. The first five axisymmetric natural frequencies are presented for uniform annular plates, of various radii and thickness ratios, with nine possible combinations of free, clamped and simply supported boundary conditions at the inner and outer edges of the plates. The accuracy of the method is established by comparing the DQM results with some exact and finite element numerical solutions and, therefore, the present DQM results could serve as a benchmark for future reference. The convergence characteristics of the method for thick plate eigenvalue problems are investigated and the versatility and simplicity of the method is established.     

Stress distribution in a linear hardening annular disk of variable thickness subjected to external pressure

The subject of investigation is a linear hardening annular disk of variable thickness subjected to external pressure. The variation of thickness is considered for two different cases: (1) h = h_o (1 − tr)^k and (2) h = h_o (1 − tr^k), where h₀, t and k are real constants. The analysis is based on Tresca's yield condition and its associated flow rule. The governing differential equations are presented in terms of radial displacement. Exact elastic—plastic solutions are obtained in terms of hypergeometric functions for the radial displacement and stress distributions. Numerical results showing the influence of the thickness variation on the stress distribution for different values of the hardening parameter are presented graphically. It can be seen from the present analysis that the thickness variation influences the radial stress and displacement distributions significantly. The present investigation may provide useful information for structural designers and engineers working in these scopes.     

Closed-form vibration analysis of thick annular functionally graded plates with integrated piezoelectric layers

This paper employs an analytical method to analyze vibration of piezoelectric coupled thick annular functionally graded plates (FGPs) subjected to different combinations of soft simply supported, hard simply supported and clamped boundary conditions at the inner and outer edges of the annular plate on the basis of the Reddy's third-order shear deformation theory (TSDT). The properties of host plate are graded in the thickness direction according to a volume fraction power-law distribution. The distribution of electric potential along the thickness direction in the piezoelectric layer is assumed as a sinusoidal function so that the Maxwell static electricity equation is approximately satisfied. The differential equations of motion are solved analytically for various boundary conditions of the plate. In this study closed-form expressions for characteristic equations, displacement components of the plate and electric potential are derived for the first time in the literature. The present analysis is validated by comparing results with those in the literature and then natural frequencies of the piezoelectric coupled annular FG plate are presented in tabular and graphical forms for different thickness-radius ratios, inner-outer radius ratios, thickness of piezoelectric, material of piezoelectric, power index and boundary conditions.     

Sustainability analyses of cutting edge radius on specific cutting energy and surface finish in side milling processes

The aim of this work is to determine the influence of cutting edge radius on the specific cutting energy and surface finish in a mechanical machining process. This was achieved by assessing the direct electrical energy demand during side milling of aluminium AW6082-T6 alloy and AISI 1018 steel in a dry cutting environment using three different cutting tool inserts. The specific energy coefficient was evaluated as an index of the sustainable milling process. The surface finish of the machined parts was also investigated after machining. It was observed that machining with the 48.50-μm cutting edge radius insert resulted in lower specific cutting energy requirements when compared with the 68.50 and 98.72-μm cutting edge radii inserts, respectively. However, as the ratio of the undeformed chip thickness to cutting edge radius is less than 1, the surface roughness increases. The surface roughness values gradually decrease as the ratio of undeformed chip thickness to cutting edge radius (h/r _e) tends to be 1 and at minimum surface roughness values when the ratio of h/r _e equalled to 1. However, the surface roughness values increased as h/r _e becomes higher than 1. This machining strategy further elucidates the black box and trade-offs of ploughing and rubbing characteristics of micro machining and optimization strategy for minimum energy and sustainable manufacture.     

Refined theory for vibrations and buckling of laminated isotropic annular plates

Hamilton's variational principle is used for the derivation of equations of transversally isotropic laminated annular plates motion. Nonlinear strain—displacements relations are considered. Linearized vibration and buckling equations are obtained for the annular plates uniformly compressed in the radial direction. The effects of transverse shear and rotational inertia are included. A closed form solution is given for the mode shapes in terms of Bessel, power and trigonometric functions. The eigenvalue equations are derived for natural frequencies and buckling loads of annular and circular plates elastically restrained against rotation along edges. Classical-type plate theory results are obtained then by letting the transverse shear stiffness go to infinity and rotational inertia go to zero. Numerical examples are presented by tables and figures for 2- and 3-layered plates with various geometrical and physical parameters. The transverse shear, rotational inertia and boundary conditions effects are discussed.     

Vibration of circular Mindlin plates with concentric elastic ring supports

This paper studies the vibration behaviour of circular Mindlin plates with multiple concentric elastic ring supports. Utilizing the domain decomposition technique, a circular plate is divided into several annular segments and one core circular segment at the locations of the elastic ring supports. The governing differential equations and the solutions of these equations are presented for the annular and circular segments based on the Mindlin-plate theory. A homogenous equation system that governs the vibration of circular Mindlin plates with elastic ring supports is derived by imposing the essential and natural boundary and segment interface conditions. The first-known exact vibration frequencies for circular Mindlin plates with multiple concentric elastic ring supports are obtained and the modal shapes of displacement fields and stress resultants for several selected cases are presented. The influence of the elastic ring support stiffness, locations, plate boundary conditions and plate thickness ratios on the vibration behaviour of circular plates is discussed.     

On the Determination of Friction Forces in Turbulent Lubrication

The mixing length theory, used previously for the determination of the velocity profiles and pressure distributions in turbulent films is now employed in order to determine the friction stresses T_o, T_h on the two lubricated surfaces and their dependence on the Reynolds number and on the pressure distribution. Thus, it results that the curves τ_o(B_xτ and –τ_h(B_x) are antisymmetric and that the derivatives τ_o/τB_x, τ_h/τB_x at B_x = 0 have constant values, independent of the Reynolds number. These results allow the determination of simple analytical formulas for the friction stresses τ_o.h as functions of the Reynolds number and of the parameter B_x, by using a linearization procedure, valid for small and moderate variations of the film thickness.

The obtained formulas are then used for the calculation of the friction forces and friction torques in journal bearings and slider bearings.     

Unsteady flow between rectangular plates and between circular plates

An analysis is presented of the unsteady flow of a viscous incompressible fluid between two parallel infinitely long rectangular plates and between two parallel circular plates. The lower plate is fixed and the upper plate moves towards the lower plate. Full Navier-Stokes equations are used to obtain the pressure distribution as a function of the film thickness h(t) and the velocity $\text{h}\text{?}\text{(t)}$ of the upper plate. The inertia is taken into account and, for a given load on the upper plate, the sinkage relation between h and t is determined for various values of the Reynolds number. The departure from the classical inertialess solution is exhibited for various values of the two dimensionless parameters involved, one characterizing the load and the other gravity.     

Numerical solution for frequencies and mode shapes of elliptical plate half of whose boundary is simply supported and the rest free

Free transverse vibrations of elliptical and circular plates of variable thickness with half of the boundary is simply supported and the rest is completely free have been computed till the first four frequencies converge to at least four significant digits. The thickness of the plate is taken varying linearly with space coordinates. Computations have been carried out by using the famous Rayleigh–Ritz method. Results are reported for various values of the taper parameters. Almost all the results reported here are entirely new. A table has been prepared to indicate the trend of convergence of the results with increasing the order of approximation. Comparison has been made with known results in special cases. For the sake of completeness, results for the cases when the entire boundary is simply supported or completely free are also reported. Three-dimensional mode shapes and the associated contour lines have been plotted in some selected cases using tools of Computer Graphics and Turbo C++.     

Exact solutions for vibration and buckling of an SS-C-SS-C rectangular plate loaded by linearly varying in-plane stresses

Exact solutions are presented for the free vibration and buckling of rectangular plates having two opposite edges (x=0 and a) simply supported and the other two (y=0 and b) clamped, with the simply supported edges subjected to a linearly varying normal stress σ_x=−N₀[1−α(y/b)]/h, where h is the plate thickness. By assuming the transverse displacement (w) to vary as sin(mπx/a), the governing partial differential equation of motion is reduced to an ordinary differential equation in y with variable coefficients, for which an exact solution is obtained as a power series (the method of Frobenius). Applying the clamped boundary conditions at y=0 and b yields the frequency determinant. Buckling loads arise as the frequencies approach zero. A careful study of the convergence of the power series is made. Buckling loads are determined for loading parameters α=0,0.5,1,1.5,2, for which α=2 is a pure in-plane bending moment. Comparisons are made with published buckling loads for α=0,1,2 obtained by the method of integration of the differential equation (α=0) or the method of energy (α=1,2). Novel results are presented for the free vibration frequencies of rectangular plates with aspect ratios a/b=0.5,1,2 subjected to three types of loadings (α=0,1,2), with load intensities N₀/N_cr=0,0.5,0.8,0.95,1, where N_cr is the critical buckling load of the plate. Contour plots of buckling and free vibration mode shapes are also shown.     

A simple method for determining the extinction ratio of strongly scattering media

In order to determine the extinction ratio k of strongly scattering media, a simple technique for instrumental elimination of the initial curvilinear part of the dependence of optical density D on medium thickness h, D(h) = D ₁ + kh, where D ₁ is the apparent optical density of the surface layers of depth h _c = 1/k (1/k is the depth of radiation penetration into the substance), is proposed. To this end, two-beam recording is used, with the sample of thickness h placed in the working cell and with the shield of thickness h ₀, made of the same material, placed in the comparison cell, provided that h _c < h ₀ < h. The extinction ratio is found from the relation k = dD/dh, where dD = D(h)-D(h ₀) is the difference of the optical densities in the working cell and in the comparison cell and dh = h-h ₀ is the difference of the thicknesses. For samples with h < h _c, two shields of thickness h ₀ are placed in each cell (in the working cell, in front of the sample and behind it) provided that h ₀ > h _c. In this case, k = dD/h, where dD = D(h + 2h ₀)-D(2h ₀).     

A new exact analytical approach for free vibration of Reissner-Mindlin functionally graded rectangular plates

An exact closed-form procedure is presented for free vibration analysis of moderately thick rectangular plates having two opposite edges simply supported (i.e. Lévy-type rectangular plates) based on the Reissner-Mindlin plate theory. The material properties change continuously through the thickness of the plate, which can vary according to a power law distribution of the volume fraction of the constituents. By introducing some new potential and auxiliary functions, the displacement fields are analytically obtained for this plate configuration. Several comparison studies with analytical and numerical techniques reported in literature are carried out to establish the high accuracy and reliability of the solutions. Comprehensive benchmark results for natural frequencies of the functionally graded (FG) rectangular plates with six different combinations of boundary conditions (i.e. SSSS-SSSC-SCSC-SCSF-SSSF-SFSF) are tabulated in dimensionless form for various values of aspect ratios, thickness to length ratios and the power law index. Due to the inherent features of the present exact closed-form solution, the present results will be a useful benchmark for evaluating the accuracy of other analytical and numerical methods, which will be developed by researchers in the future.     

Film Thickness Dependence of Lubricant Mobility

The dependence of lubricant diffusivity on lubricant film thickness is investigated by determining the best functional relationship that minimizes the error between experimental data and a numerical solution of the partial differential equation describing lubricant thickness as a function of space and time. Two optimization codes were developed. The first one is for the case of constant D _o and the second one for variable D(h). The results show that, for the case of Zdol 4000, D is a strong function of film thickness h.     

Discharge coefficient in the combined weir-gate structure

The present study investigated the flow discharge coefficient (C_dt) in the combined rectangular broad crested weir-gate structure. To this end, the effect of the following dimensionless parameters on the C_dt were investigated: the width ratio of the central weir to the width of the total structure (B/B_o), the height ratio of the central weir to the height of the central weir floor (Z/P), the ratio of the gate width to the width of the total structure (b/B_o), the ratio of the gate opening height to the height of the central weir floor (d/P), and the ratio of the head on central weir to the total head behind the structure (h₁/H). The Flow-3D numerical model, artificial intelligence models such as linear multilayer perceptron (MLP), Canfis network (CNN), recurrent network (RNN), modular neural network (MNN), and regression equation, were used to estimate the C_dt. The results indicated that increasing d/P and b/B_o ratios led to a decline in this coefficient. In the case of h₁/H ≤ 0.4, an increase in B/B_o ratio resulted in decreasing the turbulence intensity and C_dt while the impact of enhancing the size of B/B_o was not significant if h₁/H > 0.4. Besides, increasing Z/P ratio caused an increase in resistance against the flow and thus a decline in C_dt. Further, the results of artificial intelligence models and regression equation demonstrated that the MNN model with an RMSE and R² of 0.03 and 0.97, respectively, could have an accurate estimate of the C_dt values.     

A simple theory of asperity contact in elastohydro-dynamic lubrication

The Greenwood and Williamson theory of random rough surfaces in contact has been combined with established elastohydrodynamic theory to provide a theoretical approach to highly loaded lubricated contacts in which the load is shared between hydrodynamic pressure and asperity contact. It is shown that, provided a major part of the load is carried by elastohydrodynamic action, the separation between the two rough surfaces is given (to a first approximation) by the film thickness which would exist between two smooth surfaces under the same conditions of load, speed and lubricant. It then follows that the asperity pressure, both real and apparent, is determined primarily by the ratio of theoretical film thickness to the combined roughness of the two surfaces (h_o/σ). A corollary of this result is that an increase in total load, which has only a small influence on the film thickness, is carried by an increase in fluid pressure and only gives rise to a small increase in asperity contact pressure.     

Natural frequencies of symmetrically laminated elliptical and circular plates

Elliptical and circular fibre reinforced composite plates are important structural elements in modern engineering structures. Vibration analysis of these elements are of interest to structural designers. The present paper deals with the free transverse vibration analysis of symmetrically laminated solid and annular elliptic and circular plates with several complicating effects.The approach developed is based on the Rayleigh–Ritz method where the deflection of the plate is approximated by a general shape function of polynomial type.The analysis includes several complicating effects, such as the presence of an internal hole, an internal ring support, several concentrated masses and the boundary elastically restrained against rotation and translation.Several examples are solved and some results which correspond to particular cases are compared with existing values in the literature. New results are also presented for cross-ply and angle-ply elliptical and circular laminates with different boundary conditions.The algorithm developed can be applied to a wide range of elastic restraint conditions, to any aspect ratio and to higher modes. The effect of the restraint parameters along the boundary on the natural frequencies for plates with these complicating effects is considered.     

Impact damages and residual bending strength of CFRP composite laminates subjected to impact loading

The purpose of this study is to confirm the decreasing in residual bending strength, and the failure mechanisms experimentally when CFRP composite laminates are subjected to foreign object damage (FOD). Composite laminates used in this test are CFRP orthotropic laminated plates, which are stacked with two-interfaces [0^o ₆/90^o ₆] _sym and four-interfaces [0^o ₃/90^o ₆/0^o ₃] _sym . When the specimen was subjected to transverse impact by a steel ball, the delamination area generated by the impact damage was observed by using the SAM (scanning acoustic microscope). Also, the fracture surfaces obtained by three-point bending test were observed by using the SEM (scanning electron microscope). Further, failure mechanisms were investigated based on the observed delamination areas and fracture surfaces.     

Three-dimensional free vibration analysis of perforated super elliptical plates via the p-Ritz method

A three-dimensional free vibration analysis of a perforated plate with rounded corners is presented. A solution method based on the linear, small strain, three-dimensional elasticity theory and the p-Ritz algorithm is employed. This analysis method uses sets of uniquely defined one- and two-dimensional polynomial functions as the trial displacements in the thickness and surface directions to arrive the eigenvalue equation which yields the natural frequencies and mode shapes for the perforated super elliptical plates. The accuracy of these results, if possible, is validated through comparison with the available literature. Parametric investigations on the vibration behaviours of the perforated super-elliptical plates with respect to different thickness ratios, cutout sizes and boundary constraints are examined.     

Die zweiten krümmungsradien bei ebenen zwangläufenThe second curvature radii in plane kinematics

In plane Euclidean kinematics for every path the curvature radius of the evolute can be used to describe the third order properties. We investigate this second curvature radius of envelopes including the case of point-paths. For example there is a simple relation between the 2nd curvature radii of an envelope and of the path of its momentary 1st curvature center in the point of contact. In particular the derived formulas enable us to discuss the envelope c_o,if instantaneously the pole P is the 1st curvature center of the moved curve. Generally in such contact points on the pole tangent there is a self-osculation of c_o. Further we get the surprisingresult that all points ≠ P on the pole tangent have a common second curvature center, but in general distinct from that of the pole P. At last there is shown how the second curvature centers can be constructed by usual graphical methods of acceleration analysis.     

Vibration analysis of rectangular plates with edge V-notches

A method is presented for accurately determining the natural frequencies of plates having V-notches along their edges. It is based on the Ritz method and utilizes two sets of admissible functions simultaneously, which are (1) algebraic polynomials from a mathematically complete set of functions, and (2) corner functions duplicating the boundary conditions along the edges of the notch, and describing the stress singularities at its sharp vertex exactly. The method is demonstrated for free, square plates with a single V-notch. The effects of corner functions on the convergence of solutions are shown through comprehensive convergence studies. The corner functions accelerate convergence of results significantly. Accurate numerical results for free vibration frequencies and nodal patterns are tabulated for V-notched square plates having notch angle α=5° or 30° at different locations and with various notch depths. These are the first known frequency and nodal pattern results available in the published literature for rectangular plates with V-notches.